2. The set of open intervals with rational end points is equivalent in the obvious way to a subset of the set of all ordered pairs of rationals.

how is this obvious?

From what else you write this is really all we ned to show.

Let the end points of the rational interval be and with , then we introduce a map which takes the interval to the ordered pair . This is a 1-1 map of the intervals to a subset of the set of ordered pairs.

I have no clue on how to show that they map to each other, but I do have a proof showing that and are countable.

Countability of two infinite sets implies the existance of such a map between the sets.

(there are 1-1 onto maps from the naturals to each set, which are of neccesity are invertable and so the composition of one of these with the inverse of the other provides the required map between the sets)

(If you are operating with a definition of countably infinite as there being a map from N onto the set, or some other definition, you will need the proof that there is a 1-1 map that does the same job. Being rather literal minded I work with enumerable as meaning there is an enumeration which for an infinite set inplies a 1-1 onto map)